Group #13131498 — subgroups

Subgroups — #13131498

Order 1664.

440 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 198 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719629	C₂	1	2	yes	maximal
2	D₁₃×#131	C₂	1	1	yes	maximal, characteristic
3	#11719100	C₂	1	6	yes	maximal, complemented
4	D₁₃⋊#108	C₂	1	3	yes	maximal, complemented
5	C₂×#88104	C₂	1	3	yes	maximal, complemented
6	#41×D₁₃	C₂²	1	1	yes	characteristic
7	#88240	—	2	2	no	—
8	#88238	—	2	6	no	—
9	D₁₃⋊M₁₆	C₂²	1	6	yes	—
10	C₂×#6118	C₂²	1	6	yes	—
11	C₁₃×#131	C₄	1	1	yes	characteristic
12	#88202	C₄	1	1	yes	characteristic
13	#88161	C₄	1	3	yes	—
14	M₁₆×D₁₃	C₂²	1	3	yes	—
15	#88155	C₄	1	3	yes	—
16	C₂×C₈×D₁₃	C₂²	1	3	yes	—
17	#88104	C₂²	1	3	yes	—
18	#88104	C₂²	1	6	yes	—
19	#88104	C₂²	1	1	yes	characteristic
20	#88103	C₂²	1	6	yes	—
21	C₁₃×#41	C₂×C₄	1	1	yes	characteristic
22	#6131	D₄	1	1	yes	characteristic
23	Q₈×D₁₃	Q₈	1	1	yes	characteristic
24	#6129	D₄	1	3	yes	—
25	D₄×D₁₃	Q₈	1	3	yes	—
26	#6127	C₂×C₄	1	3	yes	—
27	C₂×C₄×D₁₃	C₂³	1	3	yes	—
28	#6122	—	2	6	no	—
29	C₂×#1057	—	2	6	no	—
30	#6118	—	2	6	no	—
31	#6118	C₂³	1	2	yes	—
32	#6118	C₂³	1	6	yes	—
33	D₁₃⋊C₈	—	2	6	no	—
34	C₁₃×M₁₆	C₂×C₄	1	3	yes	—
35	C₂×C₈×C₁₃	C₂×C₄	1	3	yes	—
36	#6099	C₂×C₄	1	3	yes	—
37	C₂×#1055	C₂×C₄	1	3	yes	—
38	#6094	C₂×C₄	1	6	yes	—
39	C₈×D₁₃	C₂×C₄	1	3	yes	—
40	C₈×D₁₃	C₂×C₄	1	3	yes	—
41	C₈×D₁₃	C₂³	1	3	yes	—
42	C₈×D₁₃	C₂³	1	1	yes	characteristic
43	#2916	—	13	1	no	maximal, Sylow
44	C₂²×D₁₃	C₂×Q₈	1	3	yes	—
45	Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
46	D₄×C₁₃	C₄⋊C₄	1	3	yes	—
47	C₂×C₄×C₁₃	C₂²×C₄	1	3	yes	—
48	C₂²⋊D₁₃	C₄⋊C₄	1	6	yes	—
49	C₂×Dic₁₃	C₂×D₄	1	3	yes	—
50	D₅₂	C₄⋊C₄	1	3	yes	—
51	C₄×D₁₃	C₂²×C₄	1	3	yes	—
52	C₄×D₁₃	C₂×D₄	1	3	yes	—
53	C₄×D₁₃	C₂×Q₈	1	3	yes	—
54	C₄×D₁₃	C₂⁴	1	1	yes	characteristic
55	Dic₂₆	C₄⋊C₄	1	3	yes	—
56	#1057	—	2	2	no	—
57	#1057	—	2	6	no	—
58	C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₈×C₁₃	C₂²×C₄	1	3	yes	—
60	#1055	C₂²×C₄	1	1	yes	characteristic
61	#1055	C₂²×C₄	1	3	yes	—
62	#568	—	13	2	no	—
63	C₂×#131	—	13	1	no	—
64	#430	—	13	6	no	—
65	C₂×#108	—	13	3	no	—
66	C₂×#108	—	13	3	no	—
67	C₂²×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
68	C₂×D₁₃	—	4	3	no	—
69	C₂×D₁₃	C₂×C₄⋊C₄	1	3	yes	—
70	C₂×D₁₃	C₂²×Q₈	1	1	yes	characteristic
71	C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
72	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
73	Dic₁₃	C₂×C₄⋊C₄	1	3	yes	—
74	Dic₁₃	C₂²×D₄	1	1	yes	characteristic
75	C₂×#41	—	13	1	no	—
76	#131	—	13	1	no	—
77	#131	—	13	3	no	—
78	#131	—	13	3	no	—
79	#131	—	26	6	no	—
80	#131	—	26	2	no	—
81	#131	#261	1	1	yes	characteristic
82	C₂×M₁₆	—	13	3	no	—
83	C₂×M₁₆	—	13	6	no	—
84	C₂×M₁₆	—	13	6	no	—
85	C₂²×C₈	—	13	3	no	—
86	#108	—	13	6	no	—
87	#108	—	13	3	no	—
88	#108	—	13	1	no	—
89	#108	—	13	6	no	—
90	C₂×C₁₃	—	2	3	no	—
91	C₂×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
92	D₁₃	—	2	1	no	—
93	D₁₃	—	2	3	no	—
94	#41	—	13	1	no	—
95	#41	—	13	3	no	—
96	#41	—	13	3	no	—
97	#41	C₂×#261	1	1	yes	characteristic
98	C₂×Q₈	—	13	1	no	—
99	C₂×D₄	—	13	3	no	—
100	C₂²×C₄	—	13	3	no	—
101	M₁₆	—	13	3	no	—
102	M₁₆	—	13	6	no	—
103	M₁₆	—	13	2	no	—
104	M₁₆	—	13	6	no	—
105	M₁₆	—	26	6	no	—
106	M₁₆	—	26	6	no	—
107	M₁₆	C₂×#261	1	3	yes	—
108	C₂×C₈	—	13	3	no	—
109	C₂×C₈	—	13	1	no	—
110	C₂×C₈	—	13	3	no	—
111	C₂×C₈	—	13	3	no	—
112	C₂×C₈	—	13	3	no	—
113	C₂×C₈	—	26	6	no	—
114	C₂×C₈	—	26	6	no	—
115	C₂×C₈	C₂×#261	1	3	yes	—
116	C₁₃	#2916	1	1	yes	characteristic, complemented, Sylow
117	C₂³	—	13	3	no	—
118	Q₈	—	13	3	no	—
119	Q₈	C₄⋊#261	1	1	yes	characteristic
120	D₄	—	13	6	no	—
121	D₄	—	13	3	no	—
122	D₄	C₄⋊#261	1	3	yes	—
123	C₂×C₄	—	13	3	no	—
124	C₂×C₄	—	13	3	no	—
125	C₂×C₄	—	13	3	no	—
126	C₂×C₄	—	13	1	no	—
127	C₂×C₄	—	13	3	no	—
128	C₂×C₄	C₂²×#261	1	3	yes	—
129	C₈	—	13	1	no	—
130	C₈	—	13	3	no	—
131	C₈	—	26	2	no	—
132	C₈	—	26	6	no	—
133	C₈	C₂²×#261	1	1	yes	characteristic
134	C₈	C₂²×#261	1	3	yes	—
135	C₂²	—	13	3	no	—
136	C₂²	—	13	1	no	—
137	C₂²	—	52	3	no	—
138	C₂²	C₂×C₄⋊#261	1	3	yes	—
139	C₄	—	13	3	no	—
140	C₄	—	13	1	no	—
141	C₄	C₂×C₄⋊#261	1	3	yes	—
142	C₄	C₂³×#261	1	1	yes	characteristic
143	C₂	—	2	3	no	—
144	C₂	—	26	1	no	—
145	C₂	—	26	3	no	—
146	C₂	C₂²×C₄⋊#261	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11719629

C₂

yes

maximal

D₁₃×#131

C₂

yes

maximal, characteristic

#11719100

C₂

yes

maximal, complemented

D₁₃⋊#108

C₂

yes

maximal, complemented

C₂×#88104

C₂

yes

maximal, complemented

#41×D₁₃

C₂²

yes

characteristic

#88240

—

#88238

—

D₁₃⋊M₁₆

C₂²

yes

—

C₂×#6118

C₂²

yes

—

C₁₃×#131

C₄

yes

characteristic

#88202

C₄

yes

characteristic

#88161

C₄

yes

—

M₁₆×D₁₃

C₂²

yes

—

#88155

C₄

yes

—

C₂×C₈×D₁₃

C₂²

yes

—

#88104

C₂²

yes

—

#88104

C₂²

yes

—

#88104

C₂²

yes

characteristic

#88103

C₂²

yes

—

C₁₃×#41

C₂×C₄

yes

characteristic

#6131

D₄

yes

characteristic

Q₈×D₁₃

Q₈

yes

characteristic

#6129

D₄

yes

—

D₄×D₁₃

Q₈

yes

—

#6127

C₂×C₄

yes

—

C₂×C₄×D₁₃

C₂³

yes

—

—

—

—

C₂³

yes

—

#6118

C₂³

yes

—

D₁₃⋊C₈

—

C₁₃×M₁₆

C₂×C₄

yes

—

C₂×C₈×C₁₃

C₂×C₄

yes

—

#6099

C₂×C₄

yes

—

C₂×#1055

C₂×C₄

yes

—

#6094

C₂×C₄

yes

—

C₈×D₁₃

C₂×C₄

yes

—

C₈×D₁₃

C₂×C₄

yes

—

C₈×D₁₃

C₂³

yes

—

C₈×D₁₃

C₂³

yes

characteristic

#2916

—

maximal, Sylow

C₂²×D₁₃

C₂×Q₈

yes

—

Q₈×C₁₃

C₄⋊C₄

yes

characteristic

D₄×C₁₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂²⋊D₁₃

C₄⋊C₄

yes

—

C₂×Dic₁₃

C₂×D₄

yes

—

D₅₂

C₄⋊C₄

yes

—

C₄×D₁₃

C₂²×C₄

yes

—

C₄×D₁₃

C₂×D₄

yes

—

C₄×D₁₃

C₂×Q₈

yes

—

C₄×D₁₃

C₂⁴

yes

characteristic

Dic₂₆

C₄⋊C₄

yes

—

#1057

—

#1057

—

C₈×C₁₃

C₂²×C₄

yes

characteristic

C₈×C₁₃

C₂²×C₄

yes

—

#1055

C₂²×C₄

yes

characteristic

#1055

C₂²×C₄

yes

—

—

—

—

—

—

C₂²×C₁₃

C₂×C₄⋊C₄

yes

—

C₂×D₁₃

—

C₂×D₁₃

C₂×C₄⋊C₄

yes

—

C₂×D₁₃

C₂²×Q₈

yes

characteristic

C₄×C₁₃

C₂×C₄⋊C₄

yes

—

C₄×C₁₃

C₂³×C₄

yes

characteristic

Dic₁₃

C₂×C₄⋊C₄

yes

—

Dic₁₃

C₂²×D₄

yes

characteristic

C₂×#41

—

—

—

—

—

—

yes

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

—

—

—

—

C₂×C₁₃

—

C₂×C₁₃

C₂²×C₄⋊C₄

yes

characteristic

D₁₃

—

D₁₃

—

—

—

—

yes

C₂×Q₈

—

C₂×D₄

—

100

C₂²×C₄

—

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

107

M₁₆

C₂×#261

yes

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

C₂×#261

yes

—

116

C₁₃

#2916

yes

characteristic, complemented, Sylow

117

C₂³

—

118

Q₈

—

119

Q₈

C₄⋊#261

yes

characteristic

120

D₄

—

121

D₄

—

122

D₄

C₄⋊#261

yes

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂²×#261

yes

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

C₂²×#261

yes

characteristic

134

C₈

C₂²×#261

yes

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₂×C₄⋊#261

yes

—

139

C₄

—

140

C₄

—

141

C₄

C₂×C₄⋊#261

yes

—

142

C₄

C₂³×#261

yes

characteristic

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

C₂²×C₄⋊#261

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₄⋊#261
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₄	C₂³×#261
Fitting	C₁₃×#131	C₄
socle	C₂×C₁₃	C₂²×C₄⋊C₄
prime radical	#41×D₁₃	C₂²
cosocle	C₄×D₁₃	C₂⁴
hypercenter	#131	#261
hypocenter	C₁₃	#2916
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

subgroup

G/H

center

C₂

C₂²×C₄⋊#261

commutator

C₄×C₁₃

C₂³×C₄

Frattini

C₄

C₂³×#261

Fitting

C₁₃×#131

C₄

socle

C₂×C₁₃

C₂²×C₄⋊C₄

prime radical

#41×D₁₃

C₂²

cosocle

C₄×D₁₃

C₂⁴

C₁₃

G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131

N	H ≅ G/N	type
3. #11719100	C₂	semidirect
4. D₁₃⋊#108	C₂	semidirect
5. C₂×#88104	C₂	semidirect
116. C₁₃	#2916	semidirect

H ≅ G/N

type

3. #11719100

C₂

semidirect

4. D₁₃⋊#108

C₂

semidirect

5. C₂×#88104

C₂

semidirect

116. C₁₃

#2916

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#11719629	2
D₁₃×#131	2
#11719100	2
D₁₃⋊#108	2
C₂×#88104	2
#2916	13

subgroup

index

D₁₃×#131

D₁₃⋊#108

Subgroups — #13131498

Conjugacy classes of subgroups

Complements

Maximal subgroups